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## Notes : _____ Quantum Mechanics

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**Notes : _____Quantum Mechanics**AP Physics B**Quantum?**Quantum mechanics is the study of processes which occur at the atomic scale. The word "quantum" is derived From Latin to mean BUNDLE. Therefore, we are studying the motion of objects that come in small bundles called quanta. These tiny bundles that we are referring to are electrons traveling around the nucleus.**“Newton, forgive me..”, Albert Einstein**At the atomic scale Newtonian Mechanics cannot seem to describe the motion of particles. An electron trajectory between two points for example IS NOT a perfect parabolic trajectory as Newton's Laws predicts. Where Newton's Laws end Quantum Mechanics takes over.....IN A BIG WAY! One of the most popular concepts concerning Quantum Mechanics is called , “The Photoelectric Effect”. In 1905, Albert Einstein published this theory for which he won the Nobel Prize in 1921.**What is the Photoelectric Effect?**In very basic terms, it is when electrons are released from a certain type of metal upon receiving enough energy from incident light. So basically, light comes down and strikes the metal. If the energy of the light wave is sufficient, the electron will then shoot out of the metal with some velocity and kinetic energy.**The Electron-Volt = ENERGY**Before we begin to discuss the photoelectric effect, we must introduce a new type of unit. Recall: This is a very useful unit as it shortens our calculations and allows us to stray away from using exponents.**The Photoelectric Effect**"When light strikes a material, electrons are emitted. The radiant energy supplies the work necessary to free the electrons from the surface."**Photoelectric Fact #1**The LIGHT ENERGY (E) is in the form of quanta called PHOTONS. Since light is an electromagnetic wave it has an oscillating electric field. The more intense the light the more the field oscillates. In other words, its frequency is greater.**More on Fact #1**Make sure you USE the correct constant! Planck’s Constant is the SLOPE of an Energy vs. Frequency graph!**Photoelectric Fact #2**The frequency of radiation must be above a certain value before the energy is enough. This minimum frequency required by the source of electromagnetic radiation to just liberate electrons from the metal is known as threshold frequency, f0. The threshold frequency is the X-intercept of the Energy vs. Frequency graph!**Photoelectric Fact #3**Work function, f, is defined as the least energy that must be supplied to remove a free electron from the surface of the metal, against the attractive forces of surrounding positive ions. Shown here is a PHOTOCELL. When incident light of appropriate frequency strikes the metal (cathode), the light supplies energy to the electron. The energy need to remove the electron from the surface is the WORK! Not ALL of the energy goes into work! As you can see the electron then MOVES across the GAP to the anode with a certain speed and kinetic energy.**Photoelectric Fact #4**The MAXIMUM KINETIC ENERGY is the energy difference between the MINIMUM AMOUNT of energy needed (ie. the work function) and the LIGHT ENERGY of the incident photon. THE BOTTOM LINE: Energy Conservation must still hold true! The energy NOT used to do work goes into KINETIC ENERGY as the electron LEAVES the surface. Light Energy, E WORK done to remove the electron**Putting it all together**KINETIC ENERGY can be plotted on the y axis and FREQUENCY on the x-axis. The WORK FUNCTION is the y – intercept as the THRESHOLD FREQUNECY is the x intercept. PLANCK‘S CONSTANT is the slope of the graph.**KE max = hf – hfo**• KE max = hf ( energy of the original photon ) – hfo (Work function- property of the metal) • Work function is the minimum energy needed to free an electron.**PROBLEM**• Light with frequency of 2 X 10 15 hertz is incident on a piece of copper. • A. what is the energy of light in joules and in electron volt? • If the work function for copper is 4.5 eV , what is the maximum kinetic energy , in electron volts of the emitted electrons ? • Note : 1 eV = 1.6 X 10-19 J**A. the energy in joules is given by E= hf**E = hf = (6.63X 10 -34 ) ( 2 X 1015) = E= 1.326X10-16J Since 1eV = 1.6 X 10-19 J E= 8.28 eV B. To find the maximum kinetic energy , we simply subtract the work function from the photon energy . KEmax = 8.28 V – 4.5 V = 3.79eV**Can we use this idea in a circuit?**We can then use this photoelectric effect idea to create a circuit using incident light. Of course, we now realize that the frequency of light must be of a minimum frequency for this work. Notice the + and – on the photocell itself. We recognize this as being a POTENTIAL DIFFERENCE or Voltage. This difference in voltage is represented as a GAP that the electron has to jump so that the circuit works What is the GAP or POTENTIAL DIFFERENCE is too large?**Photoelectric Fact #5 - Stopping Potential**If the voltage is TOO LARGE the electrons WILL NOT have enough energy to jump the gap. We call this VOLTAGE point the STOPPING POTENTIAL. If the voltage exceeds this value, no photons will be emitted no matter how intense. Therefore it appears that the voltage has all the control over whether the photon will be emitted and thus has kinetic energy.**Wave-Particle Duality**The results of the photoelectric effect allowed us to look at light completely different. First we have Thomas Young’s Diffraction experiment proving that light behaved as a WAVE due to constructive and destructive interference. Then we have Max Planck who allowed Einstein to build his photoelectric effect idea around the concept that light is composed of PARTICLES called quanta.**This led to new questions….**If light is a WAVE and is ALSO a particle, does that mean ALL MATTER behave as waves? That was the question that Louis de Broglie pondered. He used Einstein's famous equation to answer this question.**YOU are a matter WAVE!**Basically all matter could be said to have a momentum as it moves. The momentum however is inversely proportional to the wavelength. So since your momentum would be large normally, your wavelength would be too small to measure for any practical purposes. An electron, however, due to it’s mass, would have a very small momentum relative to a person and thus a large enough wavelength to measure thus producing measurable results. This led us to start using the Electron Microscopes rather than traditional Light microscopes.**Problem**Find the de Broglie wavelength for each of the following : • A 10g stone moving with a velocity of 20m/s • An electron (9.1 X 10-31) kg moving with a velocity of 1 X 107 m/s**since m =10g = 0.01 kg and λ = h/mv**λ = 6.63 X 10 -34 ----------------- = 3.315 X 10 -33 m (0.01)( 20) • In this part , we have λ = h / mv = 6.63 X 10 -34 ----------------- = 7.3 X 10-11 m (9.1 X 10-31)(1X 10-7)**The electron microscope**After the specimen is prepped. It is blasted by a bean of electrons. As the incident electrons strike the surface, electrons are released from the surface of the specimen. The deBroglie wavelength of these released electrons vary in wavelength which can then be converted to a signal by which a 3D picture can then be created based on the signals captured by the detector.**CW: Problems**1. What is the Broglie wavelength for a proton ( m = 1.67X 10-27 kg ) with a velocity of 6 X 10 7 m/s 2. What is the momentum associated with yellow light that has wavelength of 5,500 Å 1Å = Å X 10-10 m**1. 6.6 X 10 -15 m**• 2. 1.2 X 10 -27 kg m/s**Atomic & Nuclear Physics**AP Physics B**Objectives: After completing this module, you should be able**to: • Define and apply the concepts of mass number, atomic number, and isotopes. • Calculate the mass defect and the binding energy per nucleon for a particular isotope. • Define and apply concepts of radioactive decay and nuclear reactions. • State the various conservation laws, and discuss their application for nuclear reactions.**Life and Atoms**Every time you breathe you are taking in atoms. Oxygen atoms to be exact. These atoms react with the blood and are carried to every cell in your body for various reactions you need to survive. Likewise, every time you breathe out carbon dioxide atoms are released. The cycle here is interesting. TAKING SOMETHING IN. ALLOWING SOMETHING OUT!**The Atom**As you probably already know an atom is the building block of all matter. It has a nucleus with protons and neutrons and an electron cloud outside of the nucleus where electrons are orbiting and MOVING. Depending on the ELEMENT, the amount of electrons differs as well as the amounts of orbits surrounding the atom.**Particle**Fig. Sym Mass Charge Size Composition of Matter All of matter is composed of at least three fundamental particles (approximations): Electron e- 9.11 x 10-31 kg -1.6 x 10-19 C Proton p1.673 x 10-27 kg +1.6 x 10-19 C 3 fm Neutron n1.675 x 10-31 kg 0 3 fm The mass of the proton and neutron are close, but they are about 1840 times the mass of an electron.**Compacted nucleus:**4 protons 5 neutrons Since atom is electri-cally neutral, there must be 4 electrons. 4 electrons Beryllium Atom The Atomic Nucleus**The Bohr atom, which is sometimes shown with electrons as**planetary particles, is no longer a valid representation of an atom, but it is used here to simplify our discussion of energy levels. The uncertain position of an electron is now described as a probability distribution—loosely referred to as an electron cloud. Modern Atomic Theory**The mass number A of any element is equal to the sum of the**atomic number Z and the number of neutrons N : A = N + Z Definitions A nucleon is a general term to denote a nuclear particle - that is, either a proton or a neutron. The atomic number Zof an element is equal to the number of protons in the nucleus of that element. The mass number A of an element is equal to the total number of nucleons (protons + neutrons).**For example, consider beryllium (Be):**Symbol Notation A convenient way of describing an element is by giving its mass number and its atomic number, along with the chemical symbol for that element.**Lithium Atom**Example 1: Describe the nucleus of a lithium atom which has a mass number of 7 and an atomic number of 3. A = 7; Z = 3; N = ? N = A – Z = 7 - 3 neutrons: N = 4 Protons: Z = 3 Electrons: Same as Z**When the atom gets excited or NOT**To help visualize the atom think of it like a ladder. The bottom of the ladder is called GROUND STATE where all electrons would like to exist. If energy is ABSORBED it moves to a new rung on the ladder or ENERGY LEVEL called an EXCITED STATE. This state is AWAY from the nucleus. As energy is RELEASED the electron can relax by moving to a new energy level or rung down the ladder.**Energy Levels**Yet something interesting happens as the electron travels from energy level to energy level. If an electron is EXCITED, that means energy is ABSORBED and therefore a PHOTON is absorbed. If an electron is DE-EXCITED, that means energy is RELEASED and therefore a photon is released. We call these leaps from energy level to energy level QUANTUM LEAPS. Since a PHOTON is emitted that means that it MUST have a certain wavelength.**Energy of the Photon**We can calculate the ENERGY of the released or absorbed photon provided we know the initial and final state of the electron that jumps energy levels.**Energy Level Diagrams**To represent these transitions we can construct an ENERGY LEVEL DIAGRAM Note: It is very important to understanding that these transitions DO NOT have to occur as a single jump! It might make TWO JUMPS to get back to ground state. If that is the case, TWO photons will be emitted, each with a different wavelength and energy.**Example**An electron releases energy as it moves back to its ground state position. As a result, photons are emitted. Calculate the POSSIBLE wavelengths of the emitted photons. Notice that they give us the energy of each energy level. This will allow us to calculate the CHANGE in ENERGY that goes to the emitted photon. This particular sample will release three different wavelengths, with TWO being the visible range ( RED, VIOLET) and ONE being OUTSIDE the visible range (INFRARED)**Energy levels Application: Spectroscopy**Spectroscopy is an optical technique by which we can IDENTIFY a material based on its emission spectrum. It is heavily used in Astronomy and Remote Sensing. There are too many subcategories to mention here but the one you are probably the most familiar with are flame tests. When an electron gets excited inside a SPECIFIC ELEMENT, the electron releases a photon. This photon’s wavelength corresponds to the energy level jump and can be used to indentify the element.**Emission Line Spectra**So basically you could look at light from any element of which the electrons emit photons. If you look at the light with a diffraction grating the lines will appear as sharp spectral lines occurring at specific energies and specific wavelengths. This phenomenon allows us to analyze the atmosphere of planets or galaxies simply by looking at the light being emitted from them.**Nuclear Physics - Radioactivity**Before we begin to discuss the specifics of radioactive decay we need to be certain you understand the proper NOTATION that is used. To the left is your typical radioactive isotope. Top number = mass number = #protons + neutrons. It is represented by the letter "A“ Bottom number = atomic number = # of protons in the nucleus. It is represented by the letter "Z"**Nuclear Physics – Notation & Isotopes**An isotope is when you have the SAME ELEMENT, yet it has a different MASS. This is a result of have extra neutrons. Since Carbon is always going to be element #6, we can write Carbon in terms of its mass instead. Carbon - 12 Carbon - 14**Isotopes of helium**Helium - 3 Helium - 4 Isotopes of Elements Isotopes are atoms that have the same number of protons (Z1= Z2), but a different number of neutrons (N). (A1 A2)**A nuclide is an atom that has a definite mass numberA and**Z-number. A list of nuclides will include isotopes. The following are best described as nuclides: Nuclides Because of the existence of so many isotopes, the term element is sometimes confusing. The term nuclide is better.**Common atomic masses:**Proton: 1.007276 u Neutron: 1.008665 u Electron: 0.00055 u Atomic Mass Unit, u One atomic mass unit(1 u) is equal to one-twelfth of the mass of the most abundant form of the carbon atom--carbon-12. Atomic mass unit: 1 u = 1.6606 x 10-27 kg Hydrogen: 1.007825 u**Einstein – Energy/Mass Equivalence**In 1905, Albert Einstein publishes a 2nd major theory called the Energy-Mass Equivalence in a paper called, “Does the inertia of a body depend on its energy content?”

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